Dynamics of the angular momentum in narrow quantum rings with Rashba and Dresselhaus spin-orbit interactions

The quantum dynamics of the electron's spin and orbital angular momenta in semiconductor quantum rings is analyzed. Both Rashba and Dresselhaus spin-orbit interactions (SOIs) in their quasi-two-dimensional forms are taken into account. The narrow quantum rings are treated with models including one and two radial modes. We find that when either Rashba or Dresselhaus SOI acts alone, the different angular momentum states are coupled in blocks of two (for a single radial mode) or four (for two radial modes). We also show that the full Hilbert space splits into two disjoint subspaces, which are not coupled by either of the two SOIs, thereby decoupling accordingly the state evolution. When both SOI mechanisms are present, in principle infinitely many states are coupled, but we find by numerical computation of the quantum dynamics that for typical evolution times in practice only a few neighboring states are involved in the dynamics. Thus the exchange of angular momenta proceeds only via very few states. Furthermore, we find a trend that when initially high orbital momenta are prepared, the time evolution of spin and orbital momenta is almost unaffected by the availability of a second radial mode, in sharp contrast to the case of preparing the system in low orbital angular momentum states. The implications of our findings for the coherent control of angular momentum in quantum rings are pointed out.

Figure 1

Schematic diagram showing how the spin-orbit interactions couple the states $\left\{\right|\ell ,\sigma \rangle \}$, forming two unconnected chains. In each, every state is linked to two adjacent others with which it shares the same $\langle J_z\rangle$ or $\langle L_z-S_z\rangle$. When one interaction is turned off, the chains split into collections of independent pairs of states (two-level systems) with either definite $J_z$ (if $\beta =0$) or $L_z-S_z$ (if $\alpha =0$). The chains A and B may be constructed by applying the Hamiltonians $H_R$ and $H_D$ alternately on any state $|\ell ,\sigma \rangle$. They remain disjoint during time evolution even when both $H_R$ and $H_D$ are present.

Figure 2

Time evolution of the occupation probabilities under the action of the Rashba (left column) and Dresselhaus (right column) interactions for an electron initially in two different $H_0$ eigenstates. We consider for these calculations a GaAs QR ($m^{*}=0.063m_e$) of radius $a=50$ nm. For typical values of $\alpha$ and $\beta$, the SAM converted into OAM in one cycle is significant (nearly 50% of the initial spin polarization), and independent of the electron's energy, as higher energy ones (i.e., those of higher $\left|\ell \right|$) have the same transition amplitudes as their lower energy counterparts.

Figure 3

Maximum deviation (in units of $\hslash$) of the electron's SAM from its initial polarization $p, {max}_{0\le t<2\pi {\vartheta }_{R,D}^{-1}}|{\hslash }^{-1}\langle S_z\rangle \left(t\right)-p|$, as a function of its initial state $|\psi \rangle$, which we express parametrically as $|\psi \rangle =\sqrt{1/2+p}|\ell ,\uparrow \rangle +e^{i\varphi }\sqrt{1/2-p}|\ell +{\epsilon }_{R,D},\downarrow \rangle$, with $e^{i\varphi }$ a relative phase. The ring parameters are the same as those in Fig. 2. The centers of the purple puddles correspond to eigenstates of the two-level system, for which as expected no change in the polarization is observed. In contrast, the largest deviations ($0.7\hslash >\hslash /2$) correspond to symmetric and antisymmetric combinations of these eigenstates. The deviation observed in most cases indicates that the change induced by the SOI in the electron's polarization is macroscopic.

Figure 4

Occupation probability over time for an electron initially in $|1,\uparrow \rangle$ (top row) and $|5,\downarrow \rangle$ (bottom row), for fixed Dresselhaus and two different Rashba coupling constants. Here it is seen that the initial wave function does not spread beyond the first few neighboring states, even for large $\alpha$. Also, the extent of the spread is similar for both initial states, as suggested by Eq. (8).

Figure 5

Occupation probability over time for an electron initially in the pure state $|1,\uparrow \rangle$ of a GaAs ($m^{*}=0.063m_e$) QR of radius $a=50$ nm. Notice that the occupation of the second-nearest neighbor state $|-1,\uparrow \rangle$ reaches $30\%$ only when Rashba coupling is large enough ($\alpha /a\gtrsim E_0\approx 0.24\text{meV}$). That the optimal configuration is $\alpha =\beta$ in this case is related to the fact that transitions from $|0,\downarrow \rangle$ into and out of this state have the same probability.

Figure 6

Occupation probability over time for an electron initially in the pure state $|5,\downarrow \rangle$. The ring parameters are the same as in Fig. 5. Again, only for large Rashba couplings is there a considerable occupation of the second-nearest neighbor states, $|3,\downarrow \rangle$ and $|7,\uparrow \rangle$. In contrast to the transition frequencies, these probabilities do not differ significantly from their counterparts in the case $|1,\uparrow \rangle$ in Fig. 5. This is in line with the bound in Eq. (8), which shows that their dependence on $\ell$ cannot be too strong.

Figure 7

Ladder A of states connected by the Hamiltonian in the Shakouri et al. model. Notice that the addition of an excited radial mode in this model is completely transparent to the action of $R$, and then ladders A and B (B not shown) remain disjoint during the time evolution, as did chains A and B in the Meijer et al. model. (The states shown in this figure all gain the same global phase when rotated by $R$.) For very narrow QRs ($d/a\ll 1$), as considered by the authors (where $d/a\approx 1/8$ in [13]), inter-RM transitions are much faster than intra-RM ones. The rates of the latter are exactly the same as in the Meijer et al. model (see Sec. 3).

Figure 8

General diagram of a four-level system making up an independent block in the pure Rashba (I) and Dresselhaus (II) cases. Inter-RM transitions depend on the width of the ring and the coupling constant only [13], whereas intra-RM ones depend on $\ell$ and $\sigma$ but not on $n$ [see Eqs. (3) and (4)].

Figure 9

First two columns: Time evolution of the expectation values $\langle L_z\rangle$ and $\langle S_z\rangle$ for an electron initially in an eigenstate of $H_0, |n=0,\ell ,\uparrow \rangle$, with $\ell =0,...,5$. Each row corresponds to a different value of $\ell$. Third column: Occupation of the excited radial level relative to the ground level. Following Shakouri et al., we assume a GaAs ring of radius $a=50$ nm and a parabolic radial confinement that gives an effective ring width of $d\approx a/8$. The SOI coupling constants are $\alpha =2\beta =21.6$ meV nm.